In a sealable plastic bag (eg, ziplock) mix ½ c EVOO, lime juice, cumin and red pepper flakes. Add the chicken breasts and massage them with the mixtures. Seal the bag and refrigerate for 2 hours.
Heat ¼ c EVOO in a large (covered) skillet at medium high. Add the chicken breasts and cover. Cook in the oil for 8 minutes, turn them over and cook for another 8 minutes. Check the internal temperature (should be 165-175F). Remove to a plate. Do not drain the oil or browned bits.
Add the onions to the pan. Caramelize the onions about ¾ of the way to done, adding in the garlic about 10 minutes in. Keep covered.
Add in the pepper strips. Mix well and cook to nearly softened. Keep covered.
Add the broccoli to the top of the onion and peppers, cover and wait about 5 minutes until the broccoli is a deep green color, then mix into the rest. Add the arrowroot and mix well.
Slice the chicken breasts into ½ inch strips/slices.
After another 10 minutes or so of heating, add the chicken slices to the mixture. Add the sesame seeds. Fold the mixture well and coat the chicken with the residual liquid in the pan. Reduce the heat to a little below medium to reheat the chicken.